In practice, (projective) unitary representations of Lie groups might act on high-dimensional or even infinite-dimensional Hilbert spaces. This can be unfavourable, in particular when performing numerical simulations.
Therefore, a common practice is to instead study the (matrix) Lie group itself.
For example, an alternative for the infinite-dimensional time-evolution under a quadratic bosonic Hamiltonian (metaplectic group) can be achieved by exponentiating a corresponding finite-dimensional matrix, see sec:metaplectic.
However, a priori the distance between two unitaries cannot directly be inferred through this procedure.
Yet, this quantity is of particular interest with applications including quantum speed limits [1]}, [2]}, quantification of Trotter errors [3]} and quantum channel capacities [4]}.
Hence, the objective is to relate the distance between two unitary transformations to a suitable notion of distance on the underlying group.
In this paper, we do this by providing explicit error bounds on the distance between two unitaries \(U_g\)  and \(U_h\)  in terms of a metric \(d(g,h)\)  on the Lie group \(G\) .
